Statistical Properties of Asset Returns

Distributional properties of Dow Jones index returns

Plot the Dow Jones daily by using R built-in functions: 

## 
## Attaching package: 'psych'
## The following objects are masked from 'package:ggplot2':
## 
##     %+%, alpha
df = 3
tmp = ggplot(DJ_d, aes(sample=r_Dow_Jones/sd(r_Dow_Jones)*sqrt(df/(df-2))))
tmp + geom_qq(distribution=stats::qt, dparams=list(df=df)) + geom_abline(aes(intercept=0,slope=1),color="red")

#qqplot(rt(length(DJ_d$r_Dow_Jones),df=5),DJ_d$r_Dow_Jones)
vq = pnorm(vdata)
hist(vq)

ggplot(data=tibble(vq),aes(vq))+geom_histogram()
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

vn = NULL; for(val in grids) vn = c(vn,sum(vq <= val))
vn = c(vn[1],diff(vn))
test = sum((vn-length(vdata)/ik)**2/(length(vdata)/ik))
cat("test =",test," df =",ik-3," p-value =",1-pchisq(test,df=ik-3))
## test = 3405.87  df = 17  p-value = 0
df = 3
ndata = vdata*sqrt(df/(df-2))
vq = pt(ndata,df=5); hist(vq)

vn = NULL; for(val in grids) vn = c(vn,sum(vq <= val))
vn = c(vn[1],diff(vn))
test = sum((vn-length(vdata)/ik)**2/(length(vdata)/ik))
cat("test =",test," df =",ik-3," p-value =",1-pchisq(test,df=ik-3))
## test = 414.7555  df = 17  p-value = 0
alpha = 0.1514736
sigma = 4.0013995
ndata = vdata * sqrt((1-alpha) + alpha*sigma**2)
vq = (1-alpha)*pnorm(ndata) + alpha*pnorm(ndata,sd=sigma); hist(vq)

vn = NULL; for(val in grids) vn = c(vn,sum(vq <= val))
vn = c(vn[1],diff(vn))
test = sum((vn-length(vdata)/ik)**2/(length(vdata)/ik))
cat("test =",test," df =",ik-3," p-value =",1-pchisq(test,df=ik-3))
## test = 212.4475  df = 17  p-value = 0

Write a function for minimization to choose the optimal pair for \(\alpha\) and \(\sigma\).

func <- function(vx){
  alpha = vx[1]
  sigma = vx[2]
  ndata = vdata*sqrt((1-alpha) + alpha*sigma**2)

  vq = (1-alpha)*pnorm(ndata) + alpha*pnorm(ndata,sd=sigma)
  vn = NULL; for(val in grids) vn = c(vn,sum(vq <= val))
  vn = c(vn[1],diff(vn))
  return(sum((vn-length(vdata)/ik)**2/(length(vdata)/ik)))
}

func(c(0.15,4))
## [1] 212.7936
optim(par=c(0.15,4),fn=func,method="BFGS")
## Warning in sqrt((1 - alpha) + alpha * sigma^2): NaNs produced

## Warning in sqrt((1 - alpha) + alpha * sigma^2): NaNs produced

## Warning in sqrt((1 - alpha) + alpha * sigma^2): NaNs produced

## Warning in sqrt((1 - alpha) + alpha * sigma^2): NaNs produced

## Warning in sqrt((1 - alpha) + alpha * sigma^2): NaNs produced

## Warning in sqrt((1 - alpha) + alpha * sigma^2): NaNs produced

## Warning in sqrt((1 - alpha) + alpha * sigma^2): NaNs produced

## Warning in sqrt((1 - alpha) + alpha * sigma^2): NaNs produced
## $par
## [1] 0.1515383 4.0008180
## 
## $value
## [1] 212.4475
## 
## $counts
## function gradient 
##       54        2 
## 
## $convergence
## [1] 0
## 
## $message
## NULL
optim(par=c(0.15,4),fn=func,method="L-BFGS-B",lower=c(0,0),upper=c(1,10)) 
## $par
## [1] 0.1752619 3.7378776
## 
## $value
## [1] 190.9963
## 
## $counts
## function gradient 
##      148      148 
## 
## $convergence
## [1] 52
## 
## $message
## [1] "ERROR: ABNORMAL_TERMINATION_IN_LNSRCH"
atmp = seq(from=0.05, to=.95, length.out=40)
stmp = seq(from=1, to=10, length.out=50)
tmp = expand.grid(atmp,stmp)
res = apply(tmp,1,func)
res = t(matrix(res, nrow=length(atmp)))

library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
ret = plot_ly(x=atmp, y=stmp, z=res) %>% add_surface() %>%
  layout(scene=list(xaxis=list(title='alpha'), yaxis=list(title='sigma'), zaxis=list(title='function')))
ret

Dynamical properties of financial return series

lret = apply(log(data),2,diff)
summary(lret)
##     DAXINDX              FRCAC40           FTSE100          
##  Min.   :-0.1370990   Min.   :-0.1014   Min.   :-0.1302860  
##  1st Qu.:-0.0053976   1st Qu.:-0.0060   1st Qu.:-0.0045157  
##  Median : 0.0002581   Median : 0.0000   Median : 0.0002738  
##  Mean   : 0.0003493   Mean   : 0.0003   Mean   : 0.0004102  
##  3rd Qu.: 0.0070188   3rd Qu.: 0.0069   3rd Qu.: 0.0058174  
##  Max.   : 0.0728754   Max.   : 0.0823   Max.   : 0.0759697  
##                       NA's   :393                           
##     HNGKNGI               NIKKEI              SNGALLS          
##  Min.   :-0.4054220   Min.   :-1.614e-01   Min.   :-0.0940305  
##  1st Qu.:-0.0053347   1st Qu.:-6.014e-03   1st Qu.:-0.0041041  
##  Median : 0.0002220   Median : 0.000e+00   Median : 0.0000000  
##  Mean   : 0.0005713   Mean   : 4.991e-05   Mean   : 0.0001921  
##  3rd Qu.: 0.0079703   3rd Qu.: 6.380e-03   3rd Qu.: 0.0047405  
##  Max.   : 0.1724710   Max.   : 1.243e-01   Max.   : 0.1431300  
##                                                                
##      SPCOMP          
##  Min.   :-0.2283303  
##  1st Qu.:-0.0033884  
##  Median : 0.0003843  
##  Mean   : 0.0004885  
##  3rd Qu.: 0.0049146  
##  Max.   : 0.0870888  
## 
matplot(lret,type='l',ylab='log returns',xlab='time horizon')

sum(is.na(lret[,'FRCAC40']))
## [1] 393
LB <- function(vx,lag,ip){
  tmp = acf(vx,lag.max=lag,plot=F)$acf
  tmp = tmp[2:(lag+1)]**2
  test = sum(tmp/(length(vx)-1:lag))*length(vx)*(length(vx)+2)
  return(list(test=test, pval=1-pchisq(test,df=lag-ip)))
}

tmp = 6
LB(vx=lret[!is.na(lret[,tmp]),tmp],lag=100,ip=0)
## $test
## [1] 202.3019
## 
## $pval
## [1] 6.465107e-09